# William Thurston

August 22, 2012 4:23 PM Subscribe

"The real satisfaction from mathematics is in learning from others and sharing with others." William Thurston, one of the greatest mathematicians of the 20th century, has died. He revolutionized topology and geometry, insisting always that geometric intuition and understanding played just as important a role in mathematical discovery as did the austere formalism championed by the school of Grothendieck. Thurston's views on the relation between mathematical understanding and formal proof are summed up in his essay "On Proof and Progress in Mathematics."

In the 1970s, Thurston proposed the astonishing "geometrization conjecture," asserting that every possible 3-dimensional geometry arose by composition of elements from a short list of fundamental types (which formed the basis for a memorable Paris runway show which Thurston helped design) Geometrization formed the template for much of the progress of topology over the succeeding decades, culminating in Perelman's proof of the geometrization conjecture (and its tiny dangling corollary, the Poincare conjecture) Thurston gave an hour lecture on Perelman's proof in Paris in 2010.

Thurston has won the Fields Medal, been a director of the Mathematical Sciences Research Institute, and held faculty positions at Princeton, the University of California, and Cornell. He trained many of today's leading geometers.

In recent years, Thurston became a frequent contributor to the math Q-and-A site MathOverflow, answering many questions on subjects technical and philosophical. In answer to the question "how can I contribute to mathematics?" he wrote:

"It's not

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind...

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians."

In the 1970s, Thurston proposed the astonishing "geometrization conjecture," asserting that every possible 3-dimensional geometry arose by composition of elements from a short list of fundamental types (which formed the basis for a memorable Paris runway show which Thurston helped design) Geometrization formed the template for much of the progress of topology over the succeeding decades, culminating in Perelman's proof of the geometrization conjecture (and its tiny dangling corollary, the Poincare conjecture) Thurston gave an hour lecture on Perelman's proof in Paris in 2010.

Thurston has won the Fields Medal, been a director of the Mathematical Sciences Research Institute, and held faculty positions at Princeton, the University of California, and Cornell. He trained many of today's leading geometers.

In recent years, Thurston became a frequent contributor to the math Q-and-A site MathOverflow, answering many questions on subjects technical and philosophical. In answer to the question "how can I contribute to mathematics?" he wrote:

"It's not

*mathematics*that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind...

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians."

Terry Tao's appreciation of Thurston, including a video of a sphere turning inside out.

posted by escabeche at 4:30 PM on August 22, 2012

posted by escabeche at 4:30 PM on August 22, 2012

One more long quote, this one from "Proof and Progress":

In mathematics,it often happens that a group of mathematicians advances with a certain collection of ideas. There are theorems in the path of these advances that will almost inevitably be proven by one person or another. Sometimes the group of mathematicians can even anticipate what these theorems are likely to be. It is much harder to predict who will actually prove the theorem,although there are usually a few “point people”who are more likely to score. However, they are in a position to prove those theorems because of the collective efforts of the team.The team has a further function,in absorbing and making use of the theorems once they are proven. Even if one person could prove all the theorems in the path single-handedly,they are wasted if nobody else learns them.

There is an interesting phenomenon concerning the “point”people. It regularly happens that someone who was in the middle of a pack proves a theorem that receives wide recognition as being significant. Their status in the community—their pecking order—rises immediately and dramatically.When this happens,they usually become much more productive as a center of ideas and a source of theorems.Why? First,there is a large increase in self-esteem, and an accompanying increase in productivity. Second, when their status increases,people are more in the center of the network of ideas—others take them more seriously. Finally and perhaps most importantly, a mathematical breakthrough usually represents a new way of thinking,and effective ways of thinking can usually be applied in more than one situation.

This phenomenon convinces me that the entire mathematical community would become much more productive if we open our eyes to the real valuesin what we are doing. Jaffe and Quinn propose a system of recognized roles divided into “speculation”and “proving”. Such a division only perpetuates the myth that our progress is measured in units of standard theorems deduced. This is a bit like the fallacy of the person who makes a printout of the first 10,000 primes. What we are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding. We will be more satisfied, more productive and happier if we recognize and focus on this.

posted by escabeche at 4:32 PM on August 22, 2012

In mathematics,it often happens that a group of mathematicians advances with a certain collection of ideas. There are theorems in the path of these advances that will almost inevitably be proven by one person or another. Sometimes the group of mathematicians can even anticipate what these theorems are likely to be. It is much harder to predict who will actually prove the theorem,although there are usually a few “point people”who are more likely to score. However, they are in a position to prove those theorems because of the collective efforts of the team.The team has a further function,in absorbing and making use of the theorems once they are proven. Even if one person could prove all the theorems in the path single-handedly,they are wasted if nobody else learns them.

There is an interesting phenomenon concerning the “point”people. It regularly happens that someone who was in the middle of a pack proves a theorem that receives wide recognition as being significant. Their status in the community—their pecking order—rises immediately and dramatically.When this happens,they usually become much more productive as a center of ideas and a source of theorems.Why? First,there is a large increase in self-esteem, and an accompanying increase in productivity. Second, when their status increases,people are more in the center of the network of ideas—others take them more seriously. Finally and perhaps most importantly, a mathematical breakthrough usually represents a new way of thinking,and effective ways of thinking can usually be applied in more than one situation.

This phenomenon convinces me that the entire mathematical community would become much more productive if we open our eyes to the real valuesin what we are doing. Jaffe and Quinn propose a system of recognized roles divided into “speculation”and “proving”. Such a division only perpetuates the myth that our progress is measured in units of standard theorems deduced. This is a bit like the fallacy of the person who makes a printout of the first 10,000 primes. What we are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding. We will be more satisfied, more productive and happier if we recognize and focus on this.

posted by escabeche at 4:32 PM on August 22, 2012

Great post.

The part about geometric intuition reminds me of what my analysis prof at math camp always said when we were stuck on a problem: "Have you drawn a picture?" Drawing a picture isn't a proof, but it can often give you an intuition of what you need to do the proof.

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posted by kmz at 4:49 PM on August 22, 2012

The part about geometric intuition reminds me of what my analysis prof at math camp always said when we were stuck on a problem: "Have you drawn a picture?" Drawing a picture isn't a proof, but it can often give you an intuition of what you need to do the proof.

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posted by kmz at 4:49 PM on August 22, 2012

I believe I know just the place for that quote next week, maybe I'll find a hyperbolic skirt as well.

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posted by jeffburdges at 5:17 PM on August 22, 2012

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posted by jeffburdges at 5:17 PM on August 22, 2012

Oh no. Much too young. And a very nice write up, escabeche.

But I can't leave a memorializing dot, as that's three dimensions too few to remember Thurston by.

posted by benito.strauss at 5:25 PM on August 22, 2012 [1 favorite]

But I can't leave a memorializing dot, as that's three dimensions too few to remember Thurston by.

posted by benito.strauss at 5:25 PM on August 22, 2012 [1 favorite]

.

On another note, I can't hear "Thurston has won the Fields Medal" without hearing "

posted by thanotopsis at 5:26 PM on August 22, 2012

On another note, I can't hear "Thurston has won the Fields Medal" without hearing "

*It's not about the Fields Medal, Sean, don't you get that?!?*"posted by thanotopsis at 5:26 PM on August 22, 2012

*In the 1970s, Thurston proposed the astonishing "geometrization conjecture"...*

Actually, Thurston considered it finished work, left it for students to finish, and moved on... Perelman's work kind of settled something of a family dispute (i may be overstating this) within low-dimensional topology about the strength of Thurston's "conjecture."

posted by ennui.bz at 5:53 PM on August 22, 2012

Furthering ennui.bz, wasn't Thurston the guy who didn't publish his results? He just figured them out and told them to people, and others collected them and put them in publishable form?

posted by benito.strauss at 6:06 PM on August 22, 2012

posted by benito.strauss at 6:06 PM on August 22, 2012

A nice memorial to Thurston from Daina Taimina at Hyperbolic Crochet.

posted by escabeche at 6:14 PM on August 22, 2012

posted by escabeche at 6:14 PM on August 22, 2012

.

I met Thurston when he was a judge for the Westinghouse Science Talent Search, which I competed in. He was on a panel that interviewed contestants. (I think he was the one who, while we were being asked questions, was making sketches of us.) Talked with him briefly then--this was back in 1985.

Also, was acquainted with his son much later. He was studying some pretty advanced mathematics too.

Seems like a deep and subtle thinker, and also a passionate person. A loss.

posted by Schmucko at 6:38 PM on August 22, 2012

I met Thurston when he was a judge for the Westinghouse Science Talent Search, which I competed in. He was on a panel that interviewed contestants. (I think he was the one who, while we were being asked questions, was making sketches of us.) Talked with him briefly then--this was back in 1985.

Also, was acquainted with his son much later. He was studying some pretty advanced mathematics too.

Seems like a deep and subtle thinker, and also a passionate person. A loss.

posted by Schmucko at 6:38 PM on August 22, 2012

Not Knot (1 of 2)

+

Not Knot (2 of 2)

=

Layman's Mind-Gone Blown.

.

posted by Kinbote at 6:58 PM on August 22, 2012

+

Not Knot (2 of 2)

=

Layman's Mind-Gone Blown.

.

posted by Kinbote at 6:58 PM on August 22, 2012

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posted by matematichica at 7:10 PM on August 22, 2012

posted by matematichica at 7:10 PM on August 22, 2012

.

He was an inspiration and Three-Dimensional Geometry and Topology, Vol. 1 is beautiful.

Funny story about Thurston:

It is very, very scary to talk to Bill for extended periods of time, because you quickly develop the belief (justifiable or not) that none of your ideas, none of your intellectual accomplishments, none of your mental foundations are worth anything -- that he could have done them all in a day if he bothered.

posted by bodywithoutorgans at 7:35 PM on August 22, 2012

He was an inspiration and Three-Dimensional Geometry and Topology, Vol. 1 is beautiful.

Funny story about Thurston:

*Like John Baez on Acid*(Google Cache)It is very, very scary to talk to Bill for extended periods of time, because you quickly develop the belief (justifiable or not) that none of your ideas, none of your intellectual accomplishments, none of your mental foundations are worth anything -- that he could have done them all in a day if he bothered.

posted by bodywithoutorgans at 7:35 PM on August 22, 2012

*It is very, very scary to talk to Bill for extended periods of time, because you quickly develop the belief (justifiable or not) that none of your ideas, none of your intellectual accomplishments, none of your mental foundations are worth anything -- that he could have done them all in a day if he bothered.*

I read a poignant excerpt from a biographical or autobiographical essay recently in which Thurston was talking about his work on foliations, and how word got around that he was "hitting everything out of the park" on foliations, and how this led to other mathematicians leaving the field or not entering it in the first place, and that as a result Thurston got so lonely he stopped working on foliations himself!

posted by jamjam at 8:01 PM on August 22, 2012 [1 favorite]

.

posted by LobsterMitten at 9:49 PM on August 22, 2012

posted by LobsterMitten at 9:49 PM on August 22, 2012

.

posted by qxntpqbbbqxl at 10:18 PM on August 22, 2012

posted by qxntpqbbbqxl at 10:18 PM on August 22, 2012

*Not Knot (1 of 2)*

+

Not Knot (2 of 2)

=

Layman's Mind-Gone Blown.

+

Not Knot (2 of 2)

=

Layman's Mind-Gone Blown.

I took a couple of classes from Cameron Gordon, one of the guys who proved knot complements determine knots. Great guy.

posted by kmz at 12:10 AM on August 23, 2012

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In the math department at Berkeley in the mid-90s, there was a 3-manifold seminar where the participants would work through "Thurston's Notes", which are full of marvelous intuition but shockingly (compared to most mathematics papers) short on formal details and proofs. Most weeks, they got through about half a page. ("Yeah, we're working on the top of page 23 this week.") Occasionally they got totally stuck on a paragraph, and a small subset of the participants would promise to work through the details on their own. Sometimes, after a few weeks of effort, they would come back with a 12-page paper. Other times, after a few years of effort, they would come back with a PhD thesis.

The one thing they

posted by erniepan at 8:22 AM on August 23, 2012 [7 favorites]

In the math department at Berkeley in the mid-90s, there was a 3-manifold seminar where the participants would work through "Thurston's Notes", which are full of marvelous intuition but shockingly (compared to most mathematics papers) short on formal details and proofs. Most weeks, they got through about half a page. ("Yeah, we're working on the top of page 23 this week.") Occasionally they got totally stuck on a paragraph, and a small subset of the participants would promise to work through the details on their own. Sometimes, after a few weeks of effort, they would come back with a 12-page paper. Other times, after a few years of effort, they would come back with a PhD thesis.

The one thing they

*never*came back with was a counterexample.posted by erniepan at 8:22 AM on August 23, 2012 [7 favorites]

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posted by spinifex23 at 12:21 PM on August 23, 2012

posted by spinifex23 at 12:21 PM on August 23, 2012

.

I'm really glad he lived to see the remarkable 3-manifold-and-related stuff that's happened post-Perelman, as well. The linked works of Agol, Agol-Groves-Manning, Bergeron-Wise, Kahn-Markovic, and Wise combine to sort out Thurston's

An FPP about virtual Haken/virtual fibering would be a friendslink, and maybe not of broad interest, but perhaps someone else would like to attempt it.

posted by kengraham at 7:49 PM on August 23, 2012

I'm really glad he lived to see the remarkable 3-manifold-and-related stuff that's happened post-Perelman, as well. The linked works of Agol, Agol-Groves-Manning, Bergeron-Wise, Kahn-Markovic, and Wise combine to sort out Thurston's

**virtual fibering conjecture**, and represents a major breakthrough in low-dimensional topology that is sort of poignant in the context of Thurston's passing.An FPP about virtual Haken/virtual fibering would be a friendslink, and maybe not of broad interest, but perhaps someone else would like to attempt it.

posted by kengraham at 7:49 PM on August 23, 2012

Shattered Genius: Grigori Perelman is one of the greatest mathematicians of our time, a Russian genius who solved the Poincaré Conjecture, which plagued the brightest minds for a century. At the height of his fame, he refused a million-dollar award for his work. Then he disappeared. Our writer hunts him down on the streets of St. Petersburg.

posted by homunculus at 11:59 AM on September 1, 2012 [1 favorite]

posted by homunculus at 11:59 AM on September 1, 2012 [1 favorite]

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...insisting always that geometric intuition and understanding played just as important a role in mathematical discovery as did the austere formalism..."Ya can't have one, can't have none, you can't have one without the...o-o--other!"

posted by Mental Wimp at 4:28 PM on August 22, 2012